# Journal of Operator Theory

Volume 35, Issue 1, Winter 1996 pp. 107-115.

A parametrization of canonically Koszul invertible pairs**Authors**: Ximena P. Catepillán

**Author institution:**Department of Mathematics, Millersville University of Pennsylvania, Millersville, PA 17551, U.S.A., e-mail: xcatepil@marauder.millersv.edu

**Summary:**Let T = (T_1, T_2) be a commuting pair of operators on a Hilbert space $\mathcal H$, and let T_i = V_iP_i, i = 1, 2, be the polar decompositions of T_1 and T_2. The pair T is called canonically Koszul invertible if the Koszul complex $K(T, \mathcal H)$ admits a C*-split, i.e., if [(D^0)*]D^0]^{-1}(D^0)* and $(D^1 )*[D^1 (D^1 )*]^{-1}$ are the boundary maps of a Koszul complex, where D^0 and D^1 are the boundaries of $K(T, \mathcal H)$. We find a parametrization of canonically Koszul invertible pairs in terms of the factors V_1, P_1, V_2 and P_2. In addition, we obtain a new characterization of the commutant spectrum of T.

**Keywords:**Operators on Hilbert space, Koszul complex, commutant spectrum.

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